Estimating NOMINATE scores over time using penalized splines...PSDW-NOMINATE, our penalized spline...
Transcript of Estimating NOMINATE scores over time using penalized splines...PSDW-NOMINATE, our penalized spline...
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Estimating NOMINATE scores over timeusing penalized splines
Jeffrey B. Lewis∗ Luke Sonnet†
Department of Political ScienceUCLA
June 11, 2020‡
Abstract: DW-NOMINATE scores are the most widely-used measure of con-
gressional legislators’ positions in an abstract “ideology” space. By constrain-
ing how individual legislators’ positions can change over their careers, DW-
NOMINATE produces estimates that are comparable across time, allowing DW-
NOMINATE scores to serve as the basis of much of the research on political polar-
ization (for example Binder 2014; McCarty, Poole and Rosenthal 2006). However,
recent studies have raised concerns about the plausibility of DW-NOMINATE’s
strong constraints on member’s ideological trajectories and how those constraints∗Department of Political Science, University of California Los Angeles;
[email protected]. Corresponding author.†Department of Political Science, University of California Los Angeles;
[email protected]‡We thank seminar participants at the Stanford University GSB and University of Chicago
for their feedback on early versions of this work.
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affect inferences we make about polarization (for example Bateman and Lapinski
2016). In this paper, we develop Penalized Spline DW-NOMINATE (PSDW-
NOMINATE), a new, flexible, approach to estimating the trajectories of legis-
lators’ ideal points over time within the NOMINATE framework. We use pe-
nalized spline functions (Eilers and Marx 1996, 2010) to model each legislator’s
ideal points over her career. PSDW-NOMINATE allows us to consider a contin-
uum of degrees of constraint and to explore how the constraint that is placed on
members’ movements affects inferences about political polarization.
Word count: 11137
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1. Introduction
First introduced in 1997, DW-NOMINATE (Poole and Rosenthal 1997) scores have be-
come the most widely-used measure of the positions of members of congress in an abstract
“ideology” space. These scores are foundational to much recent empirical scholarship on the
United States congress and are central to recent studies of political polarization in the United
States.1 DW-NOMINATE is appealing largely because it uses each legislator’s entire roll call
voting record to recover efficient and objective summaries of her choices in each legislative
session that are generally highly predictive of voting on the important issues of the day and
1DW-NOMINATE scores are regularly used in news reports and commentary to describe
to the extremity of individual members of the congress (for example Aguirre 2018), shifts
in the ideologies of parties (for example Rakich 2018), or political polarization in historical
contest (for example Krugman 2016). The term “DW-NOMINATE” appears in over 850 ar-
ticles and books housed in the JSTOR digital library (https://about.jstor.org/). In the
scholarly literature, the scores are used to assess polarization (e.g. Butler 2009; Carson et al.
2007; Farina 2015; Fleisher and Bond 2004; Han and Brady 2007; Heberlig, Hetherington
and Larson 2006; Hirano et al. 2010; Krasa and Polborn 2014; McCarty, Poole and Rosenthal
2006; McTague and Pearson-Merkowitz 2013; Roberts and Smith 2003; Taylor 2003; Theri-
ault 2006), to control for, or explain, legislators’ ideological positions in studies of legislative
behavior (e.g. Binder and Maltzman 2004; Brady, Han and Pope 2007; Broz 2005; Cameron,
Kastellec and Park 2013; Carnes 2012; DeVault 2010; Gerber and Lewis 2004; Gailmard and
Jenkins 2009; Griffin and Newman 2005; Hayes 2013; Jacobsmeier 2015; Jessee and Mahotra
2010; Kleinberg and Fordham 2013; Lee, Moretti and Butler 2004; Mian, Sufi and Trebbi
2010; Rocca, Sanchez and Uscinski 2008), and to study lawmaking and gridlock (e.g. Binder
2014; Jenkins and Monroe 2012; Miller and Overby 2010; Lebo, McGlynn and Koger 2007;
Monroe and Robinson 2008; Roberts 2005; Schickler 2000) among other things.
3
https://about.jstor.org/
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which are comparable—under the maintained assumptions of the model—across time. As
described below, DW-NOMINATE places very strong and unrealistic assumptions on mem-
bers’ over-time ideological trajectories. While these strong restrictions allow members to
be placed in an way that is comparable over time, far weaker assumptions are sufficient for
that purpose. In this paper, we present a new version of DW-NOMINATE that provides for
comparable over-time measurements of members’ ideologies that also allows their ideological
positions to evolve flexibly over time.
DW-NOMINATE scores are predicated on a simple spatial model of political preference
and voting. Each legislator is assumed to have a location in an abstract ideological or
“basic” (Poole 1998) space. The space is generally taken to be two dimensional.2 Each
roll call vote is understood as a choice between two alternative positions. Legislators are
assumed to probabilistically prefer alternatives that are closer to their locations—their “ideal
points”—to alternatives that are farther away. For each roll call, a dividing line can be drawn
through the space such that those members on one side of the line are expected to vote “yea”
and those on the other are expected to vote “nay.” Empirically, legislators’ preferences, as
revealed by their roll call votes on nearly every issue that congress considers, are powerfully
summarized and predicted by their locations in this two-dimensional space.
Because members’ locations are latent quantities, the scale, location, and rotation of the
space in which they are placed is arbitrary; only the relative distances between members
are identified by the data. Over short periods of time, such as a single congress, in which
2Adding further dimensions does not greatly enhance the fit of the model (Poole and
Rosenthal 2000). Indeed, for much of US history, variation along the second dimension
accounts for little of the voting in congress.
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membership is largely static and members’ location might be taken as fixed, the arbitrary
choice of location, scale, and rotation of the two dimensions is of little consequence. However,
the question of how to anchor the space over longer periods of time as membership churns
and individual legislators’ locations may evolve is a central challenge.
Poole and Rosenthal establish the comparability of DW-NOMINATE estimates over time
by assuming that members’ positions are either fixed over time (“constant” DW-NOMINATE)
or allowed to move linearly through the space at a fixed velocity (the original DW-NOMINATE
model).3 These assumptions are restrictive and stronger than those required.
In this paper, we replace the linear trajectories assumed by DW-NOMINATE with penal-
ized smoothing splines. By adjusting the smoothing parameter of the penalized smoother,
we can continuously vary the constraint on members’ ideal points over time from the case
where member’s locations are fixed over time to the case where no constraint is imposed. We
call this new estimator Penalized Spline DW-NOMINATE, or PSDW-NOMINATE for short.
This method of modelling ideal point dynamics is isomorphic to that employed by Martin
and Quinn (2002), who model the ideologies of Supreme Court justices as a random walk
over time. The smoothing parameter that governs the constraint in our model is analogous
to the innovation variance in Martin and Quinn’s random walk.
We also address two other long-standing challenges in the implementation of DW-NOMINATE.
First, the DW-NOMINATE algorithm is computationally intensive. Our new approach uses
parallel computing to speed estimation, allowing us to apply stricter convergence criteria than
does Poole and Rosenthal’s original implementation. Second, while the original linear-change
3In the linear change variant, Poole and Rosenthal also constrain the locations of members
serving fewer than 10 years to be fixed.
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DW-NOMINATE model defined the scale of the ideological space by (largely) restricting the
space to the unit circle, it only constrains members’ locations to be inside the unit circle in
the middle of their careers. Long-serving members’ linear over-time trends can carry them
far outside of the unit circle to locations outside of the conceptual boundary of the space
that shorter-serving members cannot reach (see Figure 2 and surrounding text below) . This
strikes us as undesirable and unrealistic. We use contemporary optimizers to constrain the
ideological locations of every member in each congress to fall inside the unit-circle boundary
of the ideological space.
We apply this new estimator to all roll call votes taken from the 46th to the 115th United
States congress (1879 – 2018). We estimate the model for a range of assumptions about how
smoothly members’ ideological locations change over time.4 We then explore how dependent
conclusions about legislative polarization are on these assumptions and also explore the
distribution of over time ideological movement across members. These questions have been
considered in recent work (Caughey and Schickler 2016; Clinton, Katznelson and Lapinski
2016; Bonica 2014), but we take a different approach by embedding the possibility of flexible
ideal point dynamics directly inside the DW-NOMINATE model.
4While the optimal degree of over-time constraint can be estimated using fit criteria, as
is the case with many smoothing exercises, the choice of how much to smooth members’
locations over time is ultimately subjective. Selecting the smoothing parameter involves a
trade-off between allowing members’ full measure of preference variability to be expressed
and improving the efficiency of the estimates and the comparability of the ideological space
over time. Too much constraint averages away real changes in members’ locations. Too
little makes a consistent over-time underlying choice of location, scale and rotation for the
ideological space difficult (and, in the limit, impossible) to recover.
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We follow Poole in considering what we are measuring to be a “basic” policy space. We
will assume that members’ locations on this basic space move relatively slowly over time
and that way in which individual issues map onto this space can change over time. We
take this “basic” space to be like a price index in that it measures on-average variation that
should not be expect to reflect all changes at the level of every individual issue (or product
in the case of a price index). A more detailed discussion of what DW-NOMINATE measures
and how comparability of the scale over time is established see Appendix A. Given this
perspective, the question then becomes how best to model the trajectory of each member’s
positions over time in a way that allows changes in members’ locations to be detected while
still maintaining the comparability of the underlying ideological dimensions across time. We
present PSDW-NOMINATE as one such answer to this question.
We begin by briefly describing the workings of the DW-NOMINATE model and its as-
sumptions that establish inter-temporally comparable ideological estimates. We then present
PSDW-NOMINATE, our penalized spline approach to estimating overtime variation in a leg-
islators’ ideological locations over time. We then use our new estimator to study how varying
the level of constraint that we place on legislators’ positions across time affects the conclu-
sions that we draw about ideological change at the individual, party caucus and chamber
levels across the history of the US congress. We consider implications for our measurement
of political polarization as well as how entry and exit, when compared with within legislator
changes, contribute to political polarization. Finally, we demonstrate how our estimator
allows for within subject designs—such as difference-in-differences designs—by estimating
that presidential primary candidates from 1972–2016 move towards the “left”, regardless of
party, during the session they are vying for their party nomination.
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2. The mechanics and varieties of DW-NOMINATE
The DW-NOMINATE algorithm estimates three sets of parameters: (1) the ideal point
of each legislator i in some K-dimensional space at time t, (xit1, . . . , xitK); (2) the bill
parameters that govern the hyperplane that divides this K-dimensional space, which are the
“outcome” points associated with the yea and nay votes on roll call j at time t, (Ojkty, Ojktn);
and (3) the β and wk hyperparameters, which govern the inverse variance of the random
binary vote choices and the weight given to each of the k dimensions, respectively (Poole
2005). Here we restrict ourselves to the case where K = 2.5
The utility that the ith legislator associates with alternative a ∈ {y, n} of rollcall j in the
tth congress is defined as
Uijta = β exp(−
2∑k=1
12wk(xikt −Ojkta)
2)
+ �ijta
where �ijta i.i.d.∼ N(0,√
2/2). Accordingly, the probability of the ith legislator voting yea on
the jth roll call taken in the tth congress is Φ (Uijty − Uijtn) where Φ is the standard Normal
CDF. In order to establish the scale of the underlying space, w1 = 1 and each legislator’s
ideal point is constrained to lie in the unit circle in the middle term of their time in congress
(xit̄i1, xit̄i2). Additional constraints are placed on Ojkty and Ojktn such that at least one of
the two outcomes falls inside the unit circle and the midpoint between the two alternatives
falls inside the unit circle. The likelihood of the data given the parameters, Lnom, is simply
5In cases where K = 1, restrictions on parameters described as requirements to fall in
the unit circle become restrictions to lie in the [−1, 1] interval. For K > 2, these restrictions
become restrictions to lie in the unit (hyper)sphere.
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the product of the probabilities assigned to each vote made by each legislator over all of
observed voting decisions.
Putting aside the constraints placed on the locations of the alternatives and ideal points, it
is clear that because the utilities are functions of Euclidean distances between the ideal points
and alternatives, the same utility values obtain under any distance-preserving transformation
(such as a shift or rotation) of the ideological space. Thus, the likelihood does not uniquely
identify the parameters of the model. The various constraints outlined above help to identify
the space by establishing its scale. While full treatment of the sufficient set of restrictions
required for exact identification of DW-NOMINATE’s parameters is beyond the scope of this
paper, it should be clear that this identification problem is particularly vexing with respect
to comparing members’ locations over time. If members’ locations are allowed to vary freely
across time and if no restrictions are placed on how the votes on the same topic divide the
ideological space over time, the likelihood can be optimized congress-by-congress.6 However,
because the space recovered for each congress would then depend on an independent choice
of scale and rotation, there is no reason to suppose that the recovered locations would be
comparable from one congress to another. This problem and its consequences are discussed
in Appendix A.
In order to establish the comparability of the recovered space across time, the varieties
of DW-NOMINATE impose constraints on how members’ locations can change over time.
These constraints cause the votes taken by a member in one Congress to affect the their
estimated locations in other congresses in which they serve. Consequently, with these con-
straints, the overall likelihood cannot be broken into congress-by-congress optimizations as
6Conditional on the hyperparameters, w and β.
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long as every session has a sufficient number of returning members from the last (as is the
case in the US congress).7 Thus, these constraints overcome the incomparability problem
that arises when the likelihood can be optimized congress-by-congress. If the assumed dy-
namics are correct, it is then possible to compare the estimates from one congress to the
next and, by transitivity, each congress to any other congress.8
In the DW-NOMINATE algorithm, these three sets of parameters—the legislators’ ideal
points, the bills’ outcome points (which are a simple reparameterization of the roll call’s
cutting line), and the β, w1 and w2 hyperparameters are estimated sequentially. This means,
given the set of roll calls cast by each member, the algorithm holds the bill parameters and
hyperparameters fixed while estimating the ideal points that best explain a member’s roll
calls. Then, given those new ideal points and the same hyperparameters, the bill parameters
that best explain the set of votes cast by members on that bill are estimated. Then, the
hyperparameters are estimated given the new set of ideal points and bill parameters. This
three-step process is then repeated several times until the overall improvement in fit is small.
There are three variants of DW-NOMINATE that are commonly used today. All three set
the dimensionality, K, to two and fix the weight on the first dimension, w1, to one. The first
7Three members is sufficient in two dimensions (for rigorous treatment of this problem
in close related setting see Rivers 2003).8If a sufficient number of members serve in both the House and the Senate over their
careers and if their locations sufficiently constrained as they move from one chamber to the
other, then locations estimated for the House and Senate are also comparable. If, however,
no constraint is placed on how members’ locations change when they move from one chamber
to another then the likelihood could be optimized chamber-by-chamber and scales would not
be comparable across the chambers.
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and most commonly used set of estimates is the “common-space constant” DW-NOMINATE
ideal point scores, for which a legislator’s ideal point is held constant over their tenure.
The second, “linear change” DW-NOMINATE estimates, allow legislators to move through
the two-dimensional space over time along straight lines. The third, “Nokken-Poole” DW-
NOMINATE estimates, allow legislators to have new positions in each congress. In order to
solve the indeterminacy problem mentioned above, Nokken and Poole (2004) estimate these
congress-specific legislator ideal points by fixing the bill parameters and hyperparameters
to values estimated by the “common-space constant” algorithm. We discuss these three
commonly used varieties of DW-NOMINATE in turn.
2.1. The “common-space constant” model
In the “common-space constant” model, members of the House and Senate are jointly esti-
mated and the positions of legislators are fixed over time.9 Thus, these DW-NOMINATE
estimates only return one ideal point, (xi1, xi2), for each member and these ideal points are
constrained to fall within the unit circle that defines the DW-NOMINATE ideological space.
Note that the two-dimensional ideological space is often portrayed as an ellipse rather than
a unit circle. This is due to the second dimension weight, w2, being around half the size of
the first-dimension weight (fixed to one). When this weight is applied in the visualization
to make the importance of distances between pairs of displayed points along each of the
two dimensions comparable, the second dimension is squeezed. Furthermore, as mentioned
9An exception being where members change major parties, such as Senator Strom Thur-
mond. After a dramatic party change, these members are treated as entirely new legislators
entering congress.
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above, the bill parameters are also constrained so that one of the outcome locations, either
the yea or the nay outcome point, are within the unit circle and the midpoint between the
yea and nay outcome points falls in the unit circle. This guarantees that the line that divides
those expected to vote yea from those expected to vote nay also cuts through the unit circle.
Of course, the main shortcoming of this method is that all legislators are treated as fixed
throughout their tenures. While the assumption is sufficient to establish the comparability
of the estimated locations over time, it is far stronger than is required.
2.2. The “linear change” model
The “linear change” model relaxes the constraint that members have one ideal point through-
out their tenures and allows those that served in at least five congresses (more than 8 years)
to move in a linear fashion. Formally, fixing the locations of those serving in fewer than
five congress is sufficient to establish the comparability of the space of over time. However,
in practice and as shown below, if longer-serving members’ locations are then allowed to
vary freely, the congress-to-congress variation in the estimated locations for those members
is sufficiently large to suggest that the space is not well-established by fixing the locations
of short-serving members alone. Figure 1 shows the distribution of movements by members
of congress from 1879–2014 as estimated by the linear-change DW-NOMINATE model. The
large green bar at zero indicates the large plurality of members who served in fewer than
five congresses and are unable to move. Meanwhile, other members of congress in the right
tail of the distribution move enough to span nearly the entire DW-NOMINATE ideological
space—i.e., their total movements are near to or greater than one.
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Figure 1: Distribution of legislators by total career distance traveled in NOMINATE scoresfrom “linear change” model
House Senate
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.00
5
10
15
Distance moved
Den
sity
Congressesserved
1−4
5+
This figure plots the distribution of legislators by how much they moved along the NOMINATE1st dimension over their entire career. We take the absolute value of each session-to-session changein the “linear change” DW-NOMINATE model and sum it over each legislator’s career. Note thatwe stack those who served fewer than 5 congresses in another color as they are unable to move atall and thus their total career distance is fixed at 0.
The linear change model, while attractive because it relaxes the unrealistic constraint
that members do not change ideological positions over their tenures, has two shortcomings.
First, as implemented by Poole and Rosenthal, members are only constrained to have the
midpoint of their ideal point trajectory within the unit circle. Figure 2 shows the trajectories
of all members of congress from 1879 to 2014, estimated with the linear-change model. The
members in blue, who are in the top five percent of movement over their tenures, often
start outside or leave the circle. Second, linear movements are unlikely to capture the true
dynamics of how members’ voting behavior changes over their tenures. While the linear
change model is far less restrictive than the constant model and does establish comparable
measurements over time, it is still unnecessarily restrictive.
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Figure 2: “Linear” DW-NOMINATE positions of every member in every congress, 1879–2014
(a) House (b) Senate
Every dot represents the estimate DW-NOMINATE location of a member in given year. The dotsrepresenting the same member over time as connected by a line. The arrowhead on each line givesthe direction of movement. The blue dots and lines represent the trajectories of the fastest-moving5 percent of all Senators and House members. Notice that dots frequently appear outside of theellipse that DW-NOMINATE uses to bound the positions of the members whose locations are fixedand to bound the location of the midpoint of every member’s line of locations.
2.3. Nokken-Poole Scores
“Nokken-Poole” scores further relax constraints placed on the movements of members by
allowing them to have a separate ideal point for each congress in which they voted (See
Nokken and Poole 2004). In order to establish a comparable scale across time, Nokken
and Poole (2004) take a partial likelihood approach. To estimate members’ congress-wise
ideal points, they hold fixed the yea and nay bill parameters and hyperparameters at the
values estimated by the “common-space constant” DW-NOMINATE algorithm and then
find the optimal locations of every member in every congress resulting in distinct ideal point
estimates for each member in each congress. While not arising from an internally coherent
statistical model because the locations of the alternatives are not estimated under the same
assumptions as the members’ locations, Nokken-Poole scores do allow members’ locations to
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evolve flexibly over time. They are also useful for assessing the assumption that members’
locations are fixed across their careers by estimating how much each member’s trajectory
would deviate from constancy if allowed to do so while holding all other members’ locations
fixed over time.10
2.4. Comparing various DW-NOMINATE estimates
In Table 1, we present the correlations of the three different DW-NOMINATE scores by
party for the first NOMINATE dimension; we also include correlations with the PSDW-
NOMININATE scores that we discuss below. On the whole, the correlations among the
three canonical DW-NOMINATE models are quite high—they range from 0.82 for the corre-
lation between Nokken-Poole and the constant model for Republicans to 0.93 for the linear
change model and the constant model for the Democrats. In Table 2, we present the same
correlations but for the second NOMINATE dimension. Again, the correlations among the
different DW-NOMINATE models are quite high. Furthermore, while the correlations are
slightly higher for the Democratic party, all correlations across dimensions and parties tend
to be close to 0.9.11.
In Panel (a) of Figure 3, we present the ideal point trajectories of all legislators in the
House and Senate from the 46th congress until the 113th congress (1879–2014) for the three
10This is in fact the logic under which Nokken and Poole employ the estimator. They
consider whether the small number of legislators across US history who switch parties during
their congressional careers appear to deviate more strongly from the constancy assumption
when it is relaxed for them than do other members (Nokken and Poole 2004).11When we combine all members regardless of party, the correlations are even stronger,
with 0.96 the lowest for the first dimension and 0.86 the lowest for the second dimension
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Tabl
e1:
Cor
rela
tion
inN
OM
INAT
Efir
stdi
men
sion
acro
ssm
odel
sby
part
yD
emoc
ratic
part
y
DW
PS-D
W
Con
stan
tLi
near
Nok
ken-
Pool
eLa
mbd
a=0
Lam
bda=
500
Lam
bda=
104
Lam
bda=
107
Con
stan
tD
W1
Line
arD
W0.
931
Nok
ken-
Pool
eD
W0.
850.
921
PS-D
W(L
ambd
a=0)
0.76
0.78
0.85
1PS
-DW
(Lam
bda=
500)
0.87
0.89
0.88
0.87
1PS
-DW
(Lam
bda=
104 )
0.89
0.86
0.81
0.8
0.93
1PS
-DW
(Lam
bda=
107 )
0.91
0.83
0.76
0.76
0.86
0.95
1
Rep
ublic
anpa
rty
DW
PS-D
W
Con
stan
tLi
near
Nok
ken-
Pool
eLa
mbd
a=0
Lam
bda=
500
Lam
bda=
104
Lam
bda=
107
Con
stan
tD
W1
Line
arD
W0.
891
Nok
ken-
Pool
eD
W0.
820.
911
PS-D
W(L
ambd
a=0)
0.79
0.73
0.8
1PS
-DW
(Lam
bda=
500)
0.89
0.82
0.81
0.88
1PS
-DW
(Lam
bda=
104 )
0.92
0.81
0.77
0.82
0.96
1PS
-DW
(Lam
bda=
107 )
0.95
0.8
0.74
0.81
0.91
0.97
1
Inth
ista
ble
we
pres
ent
corr
elat
ions
bypa
rty.
Ifw
epo
olth
epa
rtie
sto
geth
eran
din
clud
ein
depe
nden
tsan
dm
embe
rsof
othe
rpa
rtie
s,ev
ery
singl
eco
rrel
atio
nis
grea
ter
.
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Tabl
e2:
Cor
rela
tion
inN
OM
INAT
Ese
cond
dim
ensio
nac
ross
mod
els
bypa
rty
Dem
ocra
ticpa
rty
DW
PS-D
W
Con
stan
tLi
near
Nok
ken-
Pool
eLa
mbd
a=0
Lam
bda=
500
Lam
bda=
104
Lam
bda=
107
Con
stan
tD
W1
Line
arD
W0.
941
Nok
ken-
Pool
eD
W0.
870.
931
PS-D
W(L
ambd
a=0)
0.85
0.84
0.89
1PS
-DW
(Lam
bda=
500)
0.91
0.86
0.82
0.89
1PS
-DW
(Lam
bda=
104 )
0.93
0.85
0.8
0.86
0.98
1PS
-DW
(Lam
bda=
107 )
0.97
0.89
0.83
0.86
0.95
0.98
1
Rep
ublic
anpa
rty
DW
PS-D
W
Con
stan
tLi
near
Nok
ken-
Pool
eLa
mbd
a=0
Lam
bda=
500
Lam
bda=
104
Lam
bda=
107
Con
stan
tD
W1
Line
arD
W0.
931
Nok
ken-
Pool
eD
W0.
820.
891
PS-D
W(L
ambd
a=0)
0.78
0.79
0.86
1PS
-DW
(Lam
bda=
500)
0.92
0.89
0.83
0.87
1PS
-DW
(Lam
bda=
104 )
0.93
0.85
0.77
0.79
0.96
1PS
-DW
(Lam
bda=
107 )
0.95
0.87
0.78
0.78
0.93
0.98
1
Inth
ista
ble
we
pres
ent
corr
elat
ions
bypa
rty.
Ifw
epo
olth
epa
rtie
sto
geth
eran
din
clud
ein
depe
nden
tsan
dm
embe
rsof
othe
rpa
rtie
s,th
em
ajor
ityof
the
corr
elat
ions
are
grea
ter,
and
noco
rrel
atio
ndi
psbe
low
0.80
.
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-
canonical DW-NOMINATE models produced by Keith Poole.12 In the first row we present
the Nokken-Poole estimates, in the second row we present the “linear change” model, and
in the third row we present the “common-space constant” model. On the y-axis is the first
dimension NOMINATE ideology score. The x-axis shows the congress. Each line in the figure
represents a single legislator’s trajectory over time. Republicans are colored red, Democrats
blue, and all independents and other parties are colored green.
There are three main insights from panel (a). First, all three show general patterns of the
Republican and Democratic parties moving away from one another in recent periods after
being relatively close in the post-World War II era. We return to this in Section 4 where we
further discuss polarization. Second, as shown in the ellipses in Figure 2, some legislators end
up outside of the unit circle in the “linear” DW-NOMINATE model (with first dimension
NOMINATE estimates greater than 1 or less than -1). Third, the Nokken-Poole estimates
are far more volatile than the other estimates. While allowing this underlying movement may
improve the fit of the model, it is unlikely that the ideology of legislators is actually moving
in this way and instead we may be overfitting the ideal points to congress-specific voting
histories. We present our new estimator, PSDW-NOMINATE to address the limitations of
these models and produce flexible, yet reasonable, estimates of legislator trajectories over
time.
12Data is archived at https://legacy.voteview.com/.
18
https://legacy.voteview.com/
-
Figu
re3:
DW
-NO
MIN
ATE
and
PSD
W-N
OM
INAT
Eid
ealp
oint
traj
ecto
ries
over
time
(a)
DW
-NO
MIN
ATE
(b)
PSD
W-N
OM
INAT
E
Each
cong
ress
isre
pres
ente
das
the
first
year
inw
hich
that
cong
ress
met
(e.g
.th
e46
thco
ngre
ssis
repr
esen
ted
as18
79).
Each
line
repr
esen
tson
em
embe
rof
cong
ress
.
19
-
3. PSDW-NOMINATE: using P-splines to model
legislators’ positions across time
We propose using P-splines (see Eilers and Marx 1996, 2010) to more reasonably and flexibly
model the trajectory of each legislator’s ideal point across her career. Rather than assume
the members of locations are fixed as in the constant DW-NOMINATE model or that their
locations track linearly as in the linear DW-NOMINATE model, we model each member’s
location using P-splines with a discrete penalty.
3.1. P-spline functions
Our objective is to provide a general, flexible and tunable method for allowing members’
locations to evolve smoothly over time that maintains the comparability of the underlying
scale. Following Poole and Rosenthal, we treat time as discrete. That is, we assume that
each legislator has a location (xi1(t), xi2(t)) for each time period t = 1, 2, . . . , Ti where Ti
represents the last congress in which member i served and t is defined such that the first
congress in which each member serves is t = 1.13 Again following Poole and Rosenthal, we
consider members’ locations to be fixed within any given congress.
In order to constrain/smooth each member’s over-time trajectories without making strong
13Note that we define t in term of the congresses since a member’s first election even if that
member’s period of service is interrupted. For example, a House member who is defeated for
reelection after her first term, returned to the House two year’s later, and then retires after
her second term in office will have Ti = 3 even though she did not serve in the congress for
which her t = 2.
20
-
parametric assumptions about those trajectories, we employ a special case of Eilers and
Marx’s 1996 penalized spline smoother. This special case was originally introduced by
Whittaker (1922) and is sometimes referred to as Whittaker smoothing. More recently,
Eilers (2003) has referred to it as “a perfect smoother.” The special case arises because we
need only evaluate each member’s location at a discrete set of evenly-spaced points in time
(t = 1, . . . , Ti). In particular, we fit a zero-degree basis for the spline with knots located at
each of the Ti periods for each member i. The choice of a zero-degree basis for the spline
is not restrictive because we need only evaluate the function at the discrete set of points at
which we have placed knots; therefore how the spline interpolates the points between knots
is irrelevant. Now we can simply write
xi1(t) = xi1t and xi2(t) = xi2t.
Note that the coefficients of the spline function are just the parameters that describe each
member’s ideal point at each period in time. So far, all we have is a long-winded way of
describing a model in which each member’s location in each congress is freely estimated as
each congress is a knot in the spline and the time between congresses is not of interest. As
discussed above, this model is not identified without further restriction. To constrain and
smooth members’ locations across time requires the imposition of the P-spline’s difference
penalties. In particular, we add an additional term to the log-likelihood, lnLnom, defined
above, that penalizes differences in the coefficients on the spline’s basis functions, which
in this case is simply a penalty on changes to each of the two coordinates describing each
21
-
legislator’s location in each congress in which they serve:
S = lnL− λ N∑i=1
Ti∑t=2
2∑k=1
(xikt − xik(t−1))2 . (1)
When λ = 0, no penalty is imposed and each legislator’s location is freely estimated in each
congress. As λ grows very large, each legislator’s ideal point is held constant across time.
At intermediate values the of λ, each member’s trajectory follows a smoothed path that falls
between the unconstrained trajectory associated with λ = 0 and the fixed location enforced
when λ = ∞. The amount of smoothness that λ dictates is a function of the curvature of
the log-likelihood. As the amount of information in the voting data increases, the effect of
the penalty diminishes. Note that we assume that the penalty is ansiotropic (moves in any
direction are penalized equally).
We hold fixed the positions of all legislators whose last congress is fewer than 10 years
since their first congress—i.e., Ti < 5. This is similar to the constraint Poole and Rosenthal
place on legislators who served fewer than 10 years, although again we allow members whose
tenures were interrupted to move so long as their first and last sessions are 10 years apart.
This constraint is important to improving the stability of estimates; without this constraint,
there is a much larger set of scales, locations, and rotations of the space that would allow
for very similar fits over time. Furthermore, when λ = 0, there is no other constraint on the
movement of legislators, and the space would be unidentified as discussed above.14
Figure 4 provides an initial illustration of how λ smooths the estimates of members’ lo-
14Note that there is nothing particularly special about having served 10 years. Other
minimum lengths of time are also possible.
22
-
cations over time. The figure shows the estimated evolution of Senator Kenneth McKellar’s
location over his 36-year career for various values of λ. A well-known Democratic Senator of
his time, McKellar represented Tennessee. Pope’s (1976) careful historical account of McKel-
lar career, relying on interviews and the documentary record, suggests that McKellar’s views
became increasingly conservative on the sorts of economic issues captured by NOMINATE’s
first-dimension as he moved away from supporting Roosevelt’s New Deal programs that he
had backed earlier in his career (Pope 1976, Chapter 6). That same historical treatment
notes that McKellar took strong segregationist positions on votes in the 1940s (Pope 1976,
Chapter 6) suggesting that his 2nd Dimension NOMINATE score might be expected to shift
during the later part of his career as well.15 We choose McKellar to illustrate how member’s
over-time trajectories are affected by choice of λ because he is among the members whose
estimated position is the most dynamic.
In sub-figure (a), we plot how McKellar moves over time along each dimension; in the
top panel we plot movement along the first dimension and in the bottom panel we plot
movement along the second dimension. In each panel, we have different lines for four different
values of λ, from 0 where movement congress-to-congress is unconstrained by the penalty, to
10,000,000 (107) where the penalty prevents almost all movement. For the values of λ where
movement is possible, we see that McKellar moves in a way consistent with Pope’s historical
account and with many Southern Democrats in the post-World War II period: towards the
center on the first dimension and and towards 1.0 on the second dimension. Along the first
dimension, we see that values of λ below 107 allow McKellar to move from around -0.3 to -0.1,
15Poole and Rosenthal (1997) observe that the second DW-NOMINATE dimension is
associated with issues of race during this time period.
23
-
as he moves towards the Republican party. Similarly we see McKellar moving towards 1.0
on the second dimension. In both panels, as λ increases, the estimated trajectory becomes
smoother and smoother. When λ is 107, there is essentially no movement, and thus McKellar
starts his career more conservatively (more positive on the first dimension) and with a higher
value on the second dimension than in the models where he is allowed to move that way
later in his career.
Sub-figure (b) plots McKellar’s trajectory along both dimensions within the NOMINATE
two-dimensional space. Each panel represents McKellar’s trajectory at a different value of λ;
the arrows indicate the new location McKellar moved to each session, and the lines become a
darker red as time passes. We can see that as λ increases, the movement towards the center
on the first dimension and movement towards 1.0 on the second dimension becomes much
smoother.
Inspection of (1) also reveals that this formulation of the P-spline is equivalent to the
placing a Gaussian random walk prior on each member’s locations over time (see Eilers,
Marx and Durbán 2015, p. 150). It is easily shown that λ is equivalent to 12∆2 where ∆2 is
the innovation variance of the equivalent Gaussian random-walk prior.16 Gaussian random
walk priors are used by Martin and Quinn (2002) to identify (establish the comparability of)
the ideological locations of US Supreme Court Justices space over time.
Given the intuitive nature of the random walk prior formulation of this approach to
smoothing and fact that the random walk priors have been used in the context of establish-
ing comparable over-time ideal point in the literature, one might wonder why we go to the
trouble constructing this estimator using the logic of P-splines. We do this for two reasons.
16The Gaussian random walk prior defines xikt ∼ N(xik(t−1),∆2).
24
-
Figure 4: Increasing lambda smooths a member’s movement over time
2nd Dimension
1st Dimension
1910 1920 1930 1940 1950
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
Year
NO
MIN
AT
E Sc
ore
Lambda=0
Lambda=500
Lambda=10^4
Lambda=10^7
(a) Each dimension over time
Lambda=10^7
Lambda=10^4
Lambda=500
Lambda=0
−1.0 −0.5 0.0 0.5 1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
DW NominateDimension 1
DW
Nom
inat
eD
imen
sion
2
(b) Movement in both dimensions
This figure shows the trajectory of Kenneth McKellar, a long-serving Democrat from Tennessee. Infigure (a), the top panel shows his movement in the first dimension over time and the bottom panelshows his movement along the second dimension over time. In figure (a), we present trajectoriesfor four different values of λ, with the color and type of the line representing the different values ofλ. Each congress is represented as the first year in which that congress met (e.g. the 62nd congressis represented as 1911). In figure (b), we plot how McKellar’s ideal point moves over time in bothdimensions at the same time. Each panel represents a different value of lambda, and the lines andarrows get darker for later years in McKellar’s career.
25
-
First, we did not want to reinterpret DW-NOMINATE as a fully Bayesian model. Given the
many complex constraints placed on the various parameters in the model (described above),
this seemed like an unnecessary challenge. Second, the P-spline logic can be extended to the
case in which members’ locations are allowed to evolve continuously through each congress
in a way that is less computationally intensive than would be the case for the random walk
prior. Bonica (2014) considers this sort of within-congress movement to members’ locations
using a very different approach and is able to address interesting questions that cannot be
answered using models that assume members’ locations are fixed within a given congress.
With the P-spline framework, a higher-order set of basis splines with a specified number
of evenly spaced knots that is much fewer than the number of legislative days could be
combined with difference penalties to compactly, yet flexibly, model legislator’s day-by-day
ideological movements. We leave this extension for future work.
In addition to the “roughness” penalty on the ideal point trajectories, we also constrain the
ideal point paths such that they can never escape the unit circle. Further, we follow Poole and
Rosenthal in constraining the parameters associated with each roll call as described above,
ensuring that cutting lines intersect the unit circle. These constraints are imposed using
inequality constraints within the SLSQP numerical optimization algorithm (Kraft 1988) as
implemented in Python’s (Python Core Team 2018) SciPy (Jones et al. 2001–) package.17
17All software and data used in this paper is available at http://github/voteview/
pynominatedyn.
26
http://github/voteview/pynominatedynhttp://github/voteview/pynominatedyn
-
3.2. Smoothing legislator’s ideal-point estimates across time
We estimate PSDW-NOMINATE using the history of roll call votes from the 46th to the
115th congress (1879 – 2018).18 We take the latest run of the common-space constant DW-
NOMINATE model on Voteview.com as our starting values19, and we treat β and w1 as
fixed, to ensure historical comparability with other NOMINATE estimates.
We also vary the size of the penalty parameter, λ, in order to study how varying constraints
on legislator movement produce different fits, legislator dynamics, and different understand-
ings of polarization. In Figure 5, we demonstrate the overall fit of the data across iterations
and the penalty parameter. The y-axis is the geometric mean probability, a convenient
monotonic transformation of the log-likehood that provides a measure of the (geometric)
average probability of the observed votes, exp lnL(V ;α,φ,λ)N
, where N is the total number of
observed of roll call vote choices. As with other regularized methods, increasing the penalty
parameter necessarily decreases the fit, as the estimator prefers simpler estimates to those
that fit the data best. Therefore, when λ is 0, the fit is highest, and when λ is 107, the fit is
lowest. Of course, better in-sample fit does not mean that the unpenalized model is the best,
nor that it accurately captures legislator dynamics. In what follows, we present results for
the unconstrained case (λ = 0), a highly constrained case (λ = 107), and two intermediate
cases, when λ = 500 and λ = 104. As mentioned above, these penalties correspond to the
“innovation” variance of a random walk, ∆2 = 12λ . The unconstrained case corresponds to
18As with the original NOMINATE models, we only admit votes where the minority of the
yeas and nays comprise at least 2.5 percent of the total vote, therefore removing unanimous
and nearly unanimous votes.19Retrieved on March 12, 2019.
27
-
Figure 5: Convergence of estimates after 30 iterations
0.74
0.75
0.76
0.77
0.78
0 10 20 30Iteration
GM
P
LambdaLambda=0
Lambda=100
Lambda=250
Lambda=500
Lambda=1000
Lambda=10^4
Lambda=10^7
Note that with greater smoothing we necessarily get poorer fits, as larger values of lambda placegreater priority on stability of estimates rather than within-congress fit.
a random-walk where the prior variance placed on changes over time is unbounded; when
λ = 500 the standard deviation of the prior on the period-to-period moves is 0.032 and when
λ = 104, the standard deviation of the prior on period-to-period moves is 0.007.
The differences in legislator movement by penalty is made clear by panel (b) of Fig-
ure 3. The y-axis is the NOMINATE first dimension ideal point; movements upwards and
downwards correspond to changes across the most important dimension of the NOMINATE
model. The first row contains the trajectories of all legislators by chamber for the uncon-
strained model. In this model, legislators vary widely congress-to-congress in ways that
are unlikely to be attributable to underlying changes in the voting decision making pro-
cess, but that fit the data quite well. Note that this model estimates trajectories with even
greater variance than the Nokken-Poole estimates in row 1 of panel (A); this is likely due to
the fact that the Nokken-Poole model takes the roll call parameters as fixed, while PSDW-
28
-
NOMINATE(λ = 0) re-estimates the roll calls and relies solely on the fixed legislators—those
whose tenures span fewer than five congresses—to fix the space. The second and third rows
of panel (b) correspond to the other two sets of PSDW-NOMINATE results we present here.
In this model legislators are able to move smoothly over time, either in a nearly linear fash-
ion, or sometimes with sharp breaks in their trajectory mid-career. The most constrained
model is nearly a constant model, although some members have such stark changes in their
voting behavior that they still move over a considerable amount of the space. The amount
of variance in the trajectories is also reflected in the total distance members move, presented
in Figure 6.
In Tables 1 and 2, we demonstrate the within-party correlation of PSDW-NOMINATE
scores for both the first and second NOMINATE dimension, respectively. As with the
DW-NOMINATE scores, the estimates are quite consistent within the PS-DWNOMINATE
models, with no correlation among them dipping below 0.76. In general, we see that the λ = 0
model has lower consistency with the other PSDW-NOMINATE estimates, largely owing
to how much congress-to-congress variance is possible in that model.20 Furthermore, the
correlations between the PSDW-NOMINATE estimates and the canonical DW-NOMINATE
estimates are also quite high, showing that the desirable properties of the flexible estimator
we present here do not come at the cost of comparability with canonical models and results
that utilize DW-NOMINATE scores.21
20When we combine all members regardless of party, the correlations with PSDW-
NOMINATE models are even stronger with are even stronger, with 0.95 the lowest correlation
for the first dimension and 0.84 the lowest for the second dimension.21Again, when we pool members across parties, the correlations between the DW-
NOMINATE and PSDW-NOMINATE estimates are even greater.
29
-
Figure 6: Distribution of legislators by total career distance traveled in NOMINATE scoresfrom penalized spline models
House Senate
Lambda=
0Lam
bda=500
Lambda=
10^4Lam
bda=10^7
0 1 2 3 0 1 2 3
0
5
10
15
0
5
10
15
0
5
10
15
0
5
10
15
Distance moved
Den
sity
Congressesspanned
1−4
5+
This figure plots the distribution of legislators by how much they moved along the NOMINATE 1stdimension over their entire career. Each represents a different λ used in PSDW-NOMINTE. Wetake the absolute value of each session-to-session change and sum it over each legislator’s career.Note that we stack those whose tenure spans fewer than 5 congresses in another color as they areunable to move at all and thus their total career distance is fixed at 0.
30
-
In Figure 7 we present the trajectories of two interesting cases; Ron Paul, Republican of
Texas (serving in the House from 1975 to 1977, 1979 to 1985, and 1997-2013), and Kirsten
Gillibrand, Democrat of New York (serving in the House from 2007 to 2009 and in the Senate
from 2009 to the present). First, with λ = 0 or λ = 500, PSDW-NOMINATE finds that
Ron Paul begins his career as a major outlier within the Republican Party, before his move
towards the rest of his party coinciding with his Republican presidential bids. However, with
larger values of λ, we see the model accommodates Paul’s later move towards the center of
the Republican party by having him start from a position closer to the other members of
the Republican party. Rather than being nearly a 1 on the right edge of the NOMINATE
first dimension, Paul starts at around a 0.75 in this constrained model. Here we see the
constrained model limiting Paul from appearing as radical in his early career, in exchange
for estimates that more easily explain his later behavior. This is emphasized in how the
unconstrained model places Paul closest to the center of the Republican party during the
two congresses in which he sought the Republican nomination for the presidency, highlighted
in the two dashed red lines.
Second, Gillibrand’s trajectory reveals similar differences between the different PSDW-
NOMINATE model estimates. In all three, Gillibrand is moving sharply to the left, especially
after her transition to the Senate in 2009. First elected to the House from a conservative
district from upstate New York, Gillibrand was seen as a fairly conservative member of
the Democratic party who became more liberal as she moved to the Senate (for example,
see Walsh, Pak and Szabo 2019). For example, she starts her congressional career with an
‘A’ rating from the National Rifle Association which has turned to an ‘F’ in recent years
(Walsh, Pak and Szabo 2019). Indeed, both of the flexible PSDW-NOMINATE models
31
-
capture this change. Gillibrand starts among the most conservative Democrats and has
moved strongly left since becoming a senator. Indeed, using PSDW-NOMINATE(λ = 500),
Gillibrand is estimated to be the most liberal member of the Senate in 2017. Again, the
more constrained PSDW-NOMINATE(λ = 104) model trades-off the difference between
her conservative days in the house with her liberal days in the Senate, and places her in
the middle of the Democratic party, slowly moving towards the left. While we cannot
definitely say which better value of λ best captures members’ true ideological trajectories,
these examples highlight how the model trades off stability against fit.
It turns out, the movement of Ron Paul and Kirsten Gillibrand is part of a systematic
movement to the “left” that legislators make when competing for their party’s nomination
for president. We explore these dynamics more fully in an applied example in Section 5.
3.3. Largest movers
In Table 3 we list the legislator’s with the greatest average session-to-session movement under
the linear DW-NOMINATE model and the PSDW-NOMINATE(λ = 500) model. For each
model, we estimate the total, weighted Euclidean distance from their final ideal point to
their initial ideal point as
∆Euc =√
(xk=1,t=T − xk=1,t=1)2 + (w2(xk=2,t=T − xk=2,t=1))2,
where T is the last session they served in, relative to the first congress they served in
(t = 1). The differences in the first and second dimensions (∆D1 and ∆D2) are measured
similarly. To compute ∆Euc we divide the total, weighted Euclidean distance by the num-
32
-
Figure 7: Trajectories of Ron Paul and Kirsten Gillibrand under different PSDW-NOMINATE models
Each congress is represented as the first year in which that congress met (e.g. the 111th congressis represented as 2009).
33
-
ber of sessions served. We then ranked all legislators by their average congress-to-congress
weighted Euclidean movement and present the top 10 from two models in Table 3. In gen-
eral, the two models agree that these members moved a great deal over their careers, with
Dubois and Schafer appearing as top ten movers under both models.
34
-
Tabl
e3:
Bigg
est
mov
ers
acro
ssdi
ffere
ntD
W-N
OM
INAT
Em
odel
sPS
-DW
(Lam
bda
=50
0)Li
near
DW
Nam
eSe
ssio
nsR
ank
∆D
1∆
D2
∆Eu
c∆
Euc
Ran
k∆
D1
∆D
2∆
Euc
∆Eu
cC
AM
PBEL
L,T
hom
asJ.
(CA
-R)
51
0.33
-0.3
60.
370.
0788
0.25
-0.2
90.
280.
06D
UBO
IS,F
red
Tho
mas
(ID
-R)
62
-0.4
20.
010.
420.
078
-0.6
9-0
.68
0.76
0.13
SCH
AFE
R,J
ohn
Cha
rles
(WI-R
)6
30.
31-0
.44
0.38
0.06
70.
01-1
.66
0.80
0.13
MA
RCA
NT
ON
IO,V
itoA
ntho
ny(N
Y-R
)7
4-0
.43
0.04
0.43
0.06
50-0
.45
-0.2
40.
460.
07PO
IND
EXT
ER,M
iles
(WA
-R)
65
0.34
-0.2
60.
360.
0620
0.51
-0.2
00.
520.
09PE
TT
IGR
EW,R
ichar
dFr
ankl
in(S
D-R
)6
6-0
.36
0.04
0.36
0.06
13-0
.54
0.30
0.56
0.09
KU
CIN
ICH
,Den
nis
(OH
-D)
87
0.13
-0.9
50.
470.
0625
1-0
.33
-0.0
60.
330.
04R
EES,
Tho
mas
Man
kell
(CA
-D)
68
0.34
0.12
0.34
0.06
159
0.29
0.04
0.29
0.05
BULO
W,W
illia
mJo
hn(S
D-D
)6
90.
33-0
.07
0.33
0.06
300.
410.
480.
470.
08BU
SBY
,Tho
mas
Jeffe
rson
(MS-
D)
610
0.33
0.01
0.33
0.05
110.
590.
460.
630.
10SM
ITH
,Fre
deric
kC
leve
land
(OH
-R)
630
40.
140.
060.
140.
021
1.01
0.13
1.01
0.17
UN
DER
HIL
L,C
harle
sLe
e(M
A-R
)6
2022
0.02
0.05
0.03
0.01
2-0
.92
0.78
1.00
0.17
JOH
NSO
N,T
imot
hyV
.(IL
-R)
626
40.
06-0
.28
0.15
0.02
30.
77-1
.30
1.00
0.17
HO
OK
,Fra
nkEu
gene
(MI-D
)5
156
-0.1
2-0
.15
0.14
0.03
4-0
.37
-1.3
70.
750.
15G
ILBE
RT,R
alph
Wal
doEm
erso
n(K
Y-D
)5
135
0.13
0.13
0.15
0.03
50.
541.
050.
740.
15BA
ILEY
,Jos
eph
Wel
don
(TX
-D)
668
-0.2
1-0
.14
0.22
0.04
6-0
.11
-1.7
90.
860.
14IN
GLI
S,R
ober
tD
urde
n(S
C-R
)6
170.
17-0
.51
0.30
0.05
90.
44-1
.07
0.68
0.11
KN
OX
,Phi
land
erC
hase
(PA
-R)
536
30.
11-0
.03
0.11
0.02
100.
520.
300.
540.
11
Thi
sta
ble
cont
ains
the
10la
rges
tm
over
sun
der
the
PSD
W-N
OM
INAT
E(λ
=50
0)an
dlin
ear
DW
-NO
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ATE
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35
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4. Application: congressional polarization
Regardless of whether and how individual member’s ideal points are allowed to move over
time, all of the variants of DW-NOMINATE allow for the study of the polarization of the
political parties by studying how much overlap there is between the parties and the distance
between averages of party members’ ideal points. A common approach is to use the mean
first dimension ideal point location of a party’s members. Figure 8 shows each member’s
location on the first dimension as well as the location of the party mean from 1879 to 2014
for the three different NOMINATE models discussed above (for example, see Poole and
Rosenthal 2017).
The three traditional NOMINATE models—(1) the fixed “common-space constant” model,
(2) the “linear change” DW-NOMINATE model, and (3) the freely moving “Nokken-Poole”
model—present a similar story about polarization in the United States congress. The party
mean locations in Figure 8 shows the parties largely converging in the early 1930s and then
diverging, with an acceleration in polarization starting in the 1970s. Furthermore, across
all three of the canonical DW-NOMINATE varieties, Republicans shifting further rightward
appear to be the main drivers of polarization. In the “common-space constant” model, the
increase in polarization can only be explained by new entrants being placed at greater ex-
tremes than their predecessors. However, when legislators are allowed to move either linearly
or by congress the resulting estimates indicate that Republican House members are becoming
more extreme through both replacement by more extreme members and through movement
away from Democrats. This can be seen by how much steeper the Republican mean in the
House is increasing in the two varieties that allow over-time movement in ideological posi-
36
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Figure 8: DW-NOMINATE estimates of party first dimension locations over time by chamberand method of estimation
Each congress is represented as the first year in which that congress met (e.g. the 46th congress isrepresented as 1879).
tions. We can also represent the growth in polarization by taking the difference between the
party means, as we do in Figure 10.
Despite weakening the assumptions made by the constant space or linear DW-NOMINATE
models, our new estimates of party means using PSDW-NOMINATE paint a similar picture
of polarization. In Figures 9 and 10, we replicate the original DW-NOMINATE figures using
our PSDW-NOMINATE estimates. In general, the implications for polarization are similar
across estimators: the distance between the two major parties has been increasing since the
end of World War II, with the gap accelerating in the 1970s and later. Furthermore, the
recent shift in polarization seems to be driven slightly more by movements in the Republican
37
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party rather than movements in the Democratic party.
There are two noticeable, although small, differences between the old NOMINATE esti-
mates and the new PSDW-NOMINATE estimates presented here. The polarization in the
late 1800s appears to be greater in the PSDW-NOMINATE models, and the House has not
become quite as polarized as the Nokken-Poole and linear change DW-NOMINATE models
suggest. In general, across both the older models and the new PSDW-NOMINATE models,
constraining legislator movement over time leads to less apparent polarization in recent years
than the more mobile models uncover. Note that this is not obvious a priori. Under a model
with no change over time whatsoever, such as the common-space constant DW-NOMINATE
model, party means may still move with the entry and exit of members. Furthermore, there
is no reason to assume that incumbents are the driving forces behind polarization, rather
than newcomers. Indeed, among the canonical DW-NOMINATE models, the constant model
suggests the least polarization in the post-World War II era.
4.1. Sources of polarization
Who is driving this observed polarization in congress? In a model where legislators are
taken as fixed over time, polarization can only occur through replacement. If no members of
congress entered or exited, party means would stay fixed. Therefore, in the common space
constant DW-NOMINATE model, only the replacement of incumbents with more extremely
positioned newcomers can drive polarization. With the PSDW-NOMINATE model, polar-
ization can come from incumbents shifting their positions as well as from replacement of
incumbents with new members with different ideologies. While this analysis is also possible
38
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Figure 9: PSDW-NOMINATE estimates of party first dimension locations over time bychamber and amount of smoothing
Each congress is represented as the first year in which that congress met (e.g. the 46th congress isrepresented as 1879).
39
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Figure 10: Combined NOMINATE estimates of inter-party polarization
Old NOMINATE PSDW−NOMINATE
House
Senate
1880
1896
1912
1928
1944
1960
1976
1992
2008
1880
1896
1912
1928
1944
1960
1976
1992
2008
0.0
0.3
0.6
0.9
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0.0
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0.6
0.9
1.2
Year
Diff
eren
ce in
1st
Dim
ensio
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rty
mea
ns
Constant
Linear
Nokken−Poole
Lambda=0
Lambda=500
Lambda=10^4
Lambda=10^7
Each congress is represented as the first year in which that congress met (e.g. the 46th congress isrepresented as 1879).
40
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with the Nokken-Poole and linear-change DW-NOMINATE models, we focus on the PSDW-
NOMINATE estimates because PSDW-NOMINATE allows members’ ideologies to evolve
over time in a way that is much less restrictive than linear-change DW-NOMINATE and
which is founded on a complete and coherent voting model unlike Nokken-Poole.
In Table 4 we present changes in party means by decade, with each panel representing
a different decade. Each row represents a set of PSDW-NOMINATE estimates at the cor-
responding value of λ. For each party, the “Total” column contains the difference in the
first dimension party from the end to the start of the decade. The “Incumbent” column
contains the difference in first dimension party mean among legislators that were present
in the the first and last congresses of the decade, and represents how much legislators who
stayed throughout the decade changed. The “Replacement” column contains the change in
first dimension party mean from the legislators who started but did not end the decade in
congress to legislators who ended but did not start the decade in congress. Therefore, this
represents the change in means among party members who were replaced—although not
necessarily in their own constituencies—over the course of a decade.
Examining the “Total” columns for both parties, we see the two parties steadily drifting
away from each other. In most decades, and across most values of λ, the Democrats move
towards -1 and the Republicans move towards +1. Only between 1979-1988 and 1999-2008
do we see the Democrats moving right, and these moves are small and only present for certain
values of λ. For the Democrats, the decades where they moved the most are 1989-1998 and
2009-2018. In both of these instances, across values of λ, Democrats move between -0.030
and -0.073 along the first dimension. When λ = 107—i.e. movement of legislators across
time is very difficult—this shift left is, by design, due to the replacement of old legislators
41
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Table 4: Change in PSDW-NOMINATE first dimension party means by decade, broken downby replacement and incumbent change
Panel A: 1979-1988Democrat Republican
Lambda Total Incumbent Replacement Total Incumbent Replacement0 -0.008 -0.023 0.014 0.086 0.060 0.110500 -0.027 -0.037 -0.012 0.076 0.050 0.100104 -0.004 -0.011 0.008 0.050 0.019 0.080107 0.006 0.000 0.017 0.038 0.000 0.075
Panel B: 1989-1998Democrat Republican
Lambda Total Incumbent Replacement Total Incumbent Replacement0 -0.073 -0.041 -0.093 0.055 0.010 0.090500 -0.066 -0.035 -0.085 0.079 0.036 0.112104 -0.039 -0.013 -0.055 0.070 0.025 0.105107 -0.030 0.000 -0.049 0.047 0.000 0.084
Panel C: 1999-2008Democrat Republican
Lambda Total Incumbent Replacement Total Incumbent Replacement0 -0.035 -0.049 -0.023 0.031 0.006 0.065500 0.023 0.016 0.022 0.018 -0.003 0.046104 -0.002 -0.009 0.001 0.036 0.009 0.072107 0.008 0.000 0.012 0.028 0.000 0.065
Panel D: 2009-2018Democrat Republican
Lambda Total Incumbent Replacement Total Incumbent Replacement0 -0.067 -0.052 -0.072 0.036 -0.002 0.065500 -0.054 -0.027 -0.073 0.049 0.016 0.075104 -0.032 -0.005 -0.057 0.045 0.009 0.072107 -0.035 0.000 -0.070 0.038 0.000 0.064
Each row represents a set of PSDW-NOMINATE estimates at the corresponding value ofλ. For each party, the “Total” column contains the difference in the first dimension partymean between the end and the start of the decade. The “Incumbent” column contains thedifference in first dimension party mean among legislators that were present in the the firstand last congresses of the decade, and represents how much legislators who stayed throughoutthe decade have changed. The “Replacement” column contains the change in first dimensionparty mean from the legislators who started but did not end the decade in congress tolegislators who ended but did not start the decade in congress.
42
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with new legislators. However, if we examine the cases where legislator movement is not
restricted—i.e. λ = 0—we see that the shift for Democrats is due to both incumbents
moving and the replacement of legislators with new Democrats. For example, if we examine
the λ = 0 estimates for the 1989 to 1998 time period, Democrats who were present at the
beginning and end of the decade moved 0.041 points to the left, and legislators who left were
replaced with new legislators who were on average 0.093 points to the left. In almost all
specifications and decades where Democrats move a considerable amount, it is due to both
incumbents changing and replacement, although the replacement effect appears to introduce
slightly more left-leaning legislators.
For the Republican party, the estimates depict a very similar pattern, although the picture
is more consistent across decades and models. In all decades and for all specifications, the
Republican party moved towards the right. Furthermore, for all decades and all specifica-
tions, the mean among replaced legislators moves more than the mean among legislators who
served the full decade. Furthermore, the change among incumbent Republicans goes to zero
in the later decades, indicating that the vast majority of the Republican contribution to po-
larization comes from the replacement of less conservative legislators with more conservative
legislators.
Table 4 also demonstrates how increasing the smoothing parameter λ changes our un-
derstanding of the determinants of polarization. For both parties across all decades, when
λ = 107, there is no movement in incumbent legislator’s positions by design as the estimates
are all 0.000. Thus in these models, all polarization is due to replacement. However, even
when we relax the smoothing parameter and allow incumbents to move, replacement remains
a large driver of polarization.
43
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5. Application: legislative behavior of presidential primary
candidates
Allowing legislators to move over time also permits within-legislator designs, such as difference-
in-differences designs that study, for example, the effects of reelection campaigns or geographically-
clustered exogenous shocks. Of course, smoothing the splines suppress over-time variation
resulting in conservative estimated effects. However, in the example presented below, there
is little evidence that the smoothing spline substantially attenuates the estimated effects
except at the largest values of λ.
In this section, we use a difference-in-differences design to examine how the legislative
behavior of candidates competing for their party’s presidential nomination changes during
the session they are running for president. We use data from the America Votes series
published by CQ Press to code all presidential primary candidates that appeared on any
state primary ballot from the 1972 through 2012 elections for the Republican and Democratic
parties (Scammon, McGillivray and Cook N.d.). We manually coded those who appeared
on 2016 election state ballots. We then manually checked which of those candidates were
serving in the U.S. Congress at any point in the session during which the primary and general
elections were held. We then subset the PSDW-NOMINATE data to the sessions that span
from the 1972 election to the 2016 election and we only admit legislators who served at least
five sessions of congress and thus are permitted to move by our estimator. Using this sample,
we estimate the difference-in-differences model
Yij = τPij + Legislatori + Sessionj,
44
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where Yij is the first dimension PSDW-NOMINATE score for the ith legislator22 in session
j, Pij is that legislator-session’s indicator for whether the corresponding legislator appeared
on a major party’s primary ballot during that session, and Legislatori and sessionj are the
unit (legislator) and time (session) fixed effects.
We fit this model using OLS, estimate heteroskedastic consistent (HC1) standard errors,
and present the results for four different values of λ in Table 5.23 We first split the sample
into Democrats and Republicans, and report the results for each sample in Panels A and
B, respnectively.24 Each column represents a separate difference-in-differences model esti-
mated using a different set of PS-DWNOMINATE(λ) estimates. For all four values of λ
and across both parties, presidential primary candidates move towards the “left”—i.e. their
first-dimension NOMINATE score becomes more negative—during the session they are on
presidential primary ballots, relative to the rest of their career and the movements of other
legislators in the same sessions. For Democrats, in Panel A, this estimate is only statistically
significant at the 0.05 level for PSDW-NOMINATE(λ = 500), while for the Republicans the
estimates are all statistically significant at the 0.05 level and are substantially larger than the
estimates for Democrats. The magnitude of the effect is also relatively large; the standard
deviation of PSDW-NOMINATE(λ = 500) 1st dimension scores is about 0.12 for both par-
ties in the 2015-2016 session. The estimates in column 2 indicate that legislators move to the
“left” about one-third to one-half of the standard deviation of their party’s DW-NOMINATE
22Technically we use ICPSR number, as discussed above, where legislator’s who change
party are considered separate legislator’s for the sake of DW-NOMINATE estimation.23We also present analogous clustered standard errors at the legislator level in braces.24Bernie Sanders is not officially a Democrat, however as he was seeking the Democratic
nomination for president, he is included in the Democrats sample.
45
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scores during a presidential campaign.
Table 5: Difference-in-differences estimates of legislator first dimension estimates duringpresidential primary campaigns
Outcome: DW Nominate, 1st Dimension
Panel A: DemocratsLambda=0 Lambda=500 Lambda=104 Lambda=107
Pres. Primary Candidate -0.014 -0.035∗ -0.011† -0.000(0.021) (0.014) (0.006) (0.000)[0.021] [0.014] [0.007] [0.000]
R2 0.695 0.857 0.948 1.000Num. obs. 5454 5454 5454 5454
Panel B: RepublicansLambda=0 Lambda=500 Lambda=104 Lambda=107
Pres. Primary Candidate -0.061∗ -0.056∗ -0.022∗ -0.000∗(0.031) (0.025) (0.010) (0.000)[0.044] [0.037] [0.013] [0.000]
R2 0.796 0.911 0.976 1.000Num. obs. 4293 4293 4293 4293
∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05; †p < 0.1. All models fit using OLS with legislator andsession fixed effects. Heteroskedastic consistent (HC1) standard errors in parentheses andare used for the significance tests reported using the stars. Analogous clustered standarderrors (CR1), with clustering by legislator, are reported in brackets. Data used are PSDW-Nominate estimates subset to the 92nd-114th congresses. Only legislators who served morethan four sessions are admitted in to this analysis. See Appendix B for the same model withthose legislators added back in, as well as for estimates when pooling across party.
Of course, the magnitude of this estimate for the larger values of λ is much smaller than for
the smaller values of λ as the within-legislator variation decreases. This is also represented
in the R2—it approaches 1.00 as λ increases and the individual fixed effects explain nearly
all of the variation in NOMINATE scores. Nonetheless, the value of λ primarily will affect
the scale of any movements—the rollcalls and what they imply for legislator ideal points
are unchanged. Therefore, even with a large λ we should expect similarly signed, although
46
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smaller scale, movements.
Because there are few presidential candidates in this time frame who are also legislators,
we plot each of their individual PSDW-NOMINATE 1st Dimension trajectories in Figure 11.
Each panel represents one legislator in our difference-in-differences sample who appeared on
a major party’s primary ballot at least once. The x-axis is the second year of a session of
congress, centered at each legislator’s first primary run. Therefore, someone running for the
first time in 2012 would have the session ending in 2012 as 0, and then the subsequent session
would be coded as 2 instead of 2014. The y-axis is the PSDW-NOMINATE 1st dimension
ideal estimate centered to the 1st dimension estimate from the session of their first primary
bid. Therefore, each panel shows the legislator’s 1st dimension position relative to their first
presidential campaign. We also highlight all presidential campaigns with black dots, and we
plot the trajectories for the two more volatile values of λ. The panels are ordered by average
rescaled PSDW-NOMINATE 1st Dimension scores, showing that Bernie Sanders and Ron
Paul have the highest average 1st dimension scores over their careers, relative to the year
they first ran for president.
6. Conclusion
By taking a modern and flexible approach to modelling members’ ideal-point dynamics, the
PSDW-NOMINATE model relaxes the strong assumptions of the original linear-change and
constant-space DW-NOMINATE models. While estimates produced by PSDW-NOMINATE
are highly correlated with those of the original estimators, the new estimators allow re-
searchers to study and account for over-time changes in members’ locations in ways that
47
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had heretofore not been possible.
Interestingly, allowing for more flexible dynamics in members’ locations does not by-and-
large alter Poole’s (2007) conclusion that most members’ locations vary little over time.
While we highlighted a few members whose locations did evolve substantially over their
careers and we could (ex post) square those changes with the historical and qualitative
accounts of those members’ ideological evolution, we leave to future work the careful study
of which members move, by how much, and why.
Given that so much of our understanding of polarization in the contemporary US congress
and the nation more generally relies on DW-NOMINATE estimates, it is reassuring that
those results are generally reproduced by PSDW-NOMINATE. Consistent with the previ-
ous DW-NOMINATE estimates, we find that it is replacement rather than the ideological
drift of continuing members that accounts for the bulk of the increase in partisan polariza-
tion observed in recent decades. Neither the strong assumptions about member’s over-time
trajectories or the estimation compromises arising from the computing-limitations faced by
Poole and Rosenthal in the 1990s drive the conclusion that the congress is more polarized
now than it has been at any point since the 19th Century.
Moving forward, we expect and intend that PSDW-NOMINATE estimates will supplant
the ageing DW-NOMINATE scores as the core summary of members’ general ideological
dispositions and roll call voting records.
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Figure 11: Mean PSDW-NOMINATE 1st dimension trajectories for presidential primarycandidates
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